The graph of a sinusoidal function has a maximum point at $(0,7)$ and then intersects its midline at $(3,3)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline intersection is at $y={3}$, so this is the midline. The maximum point is $4$ units above the midline, so the amplitude is ${4}$. The midline intersection is $3$ units to the right of the maximum point, so the period is $4\cdot 3={12}$. [Why did we multiply by 4?] Determining the type of function to use Since the graph has an extremum point at $x=0$, we should use the cosine function and not the sine function. This means there's no horizontal shift, so the function is of the form $a\cos(bx)+d$. [How do we know that?] Determining the parameters in $a\cos(bx)+d$ Since the extremum point at $x=0$ is a maximum point, we know that $a>0$. [How do we know that?] The amplitude is ${4}$, so $|a|={4}$. Since $a>0$, we can conclude that $a=4$. The midline is $y={3}$, so $d=3$. The period is ${12}$, so $b=\dfrac{2\pi}{{12}}=\dfrac{\pi}{6}$. The answer $f(x)=4\cos\left(\dfrac{\pi}{6}x\right)+3$